Question: Compute
\[\sum_{n = 1}^\infty \frac{1}{n(n + 2)}.\]
Explanation: By partial fractions,
\[\frac{1}{n(n + 2)} = \frac{1/2}{n} - \frac{1/2}{n + 2}.\]Hence,
\begin{align*}
\sum_{n = 1}^\infty \frac{1}{n(n + 2)} &= \left( \frac{1/2}{1} - \frac{1/2}{3} \right) + \left( \frac{1/2}{2} - \frac{1/2}{4} \right) + \left( \frac{1/2}{3} - \frac{1/2}{5} \right) + \left( \frac{1/2}{4} - \frac{1/2}{6} \right) + \dotsb \\
&= \frac{1/2}{1} + \frac{1/2}{2} \\
&= \boxed{\frac{3}{4}}.
\end{align*}